Zero

Zero (0) is a that number which describes the size of an empty set, or nothing. It is of crucial importance in [positional number systems]?, where the position of a digit signifies its value, and the digit zero is used to skip a position.

Zero was introduced into mathematics relatively late, during the early 800s in Indian books. The definition originally used was, "In subtraction, when nothing is left over, write a dot to keep the place." Prior mathematical systems, such as the Babylonian or Greek, did not use zero at all, but still succeded quite well for everyday purposes.

The following are some basic rules for dealing with zero. These rules apply for any complex number x, unless otherwise stated.

• Addition: x + 0 = x and 0 + x = x. (That is, 0 is an identity element with respect to addition.)
• Subtraction: x - 0 = x and 0 - x = -x.
• Multiplication: x × 0 = 0 and 0 × x = 0.
• Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse.
• Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts. For all positive x, 0^x = 0.

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The year zero does not exist. Instead there is a "zero point" in time between the years [1 B.C.]? and 1.